I am in the following situation. I have a stable presentable $\infty$-category $\cal{C}$, and a sequence of full stable subcategories $\dots\subset\cal{C}_{-2}\subset\cal{C}_{-1}\subset\cal{C}_0\subset\cal{C}_1\subset\dots\subset\cal{C}$ such that $\cap_i\cal{C}_i=0$ and $\text{colim}_i\ \cal{C}_i=\cal{C}$. We can then consider $\lim_i\cal{C}/\cal{C}_i$ suitably interpreted. I want to know if there are (nontrivial) results out there relating $K_0(\cal{C})$ and $K_0(\lim_i\cal{C}/\cal{C}_i)$, possibly for concrete $\cal{C}$. In particular, I want to see cases studying the extent to which $K_0(\cal{C})\rightarrow K_0(\lim_i\cal{C}/\cal{C}_i)$ is injective. Also, are there results relating $K_0(\lim_i\cal{C}/\cal{C}_i)$ to $\lim_iK_0(\cal{C}/\cal{C}_i)$?
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